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\begin{document}

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\section*{Toda bracket}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-topos theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[abelian categories]] one talks of [[chain complexes]]; and in that context a composable pair $A \to B \to C$ is null iff $B \to C$ factors through the [[cokernel]] $B/(A)$:

\begin{displaymath}
\begin{array}{ccccc}
A & \to & B \\
\downarrow &  & \downarrow & \searrow \\
0 & \to & B/(A) & \to & C
\end{array}
\end{displaymath}
and so forth. In a strict context, the factorization is unique.

In a pointed [[(∞,1)-category]] with [[(∞,1)-colimits]] of small [[1-truncated]] [[diagrams]], one may still consider factorizations through [[cofibers]]: $A \to B \to C \sim * : A \to C$ but now there is a choice to make, roughly parametrized by an [[action]] of $Map_* (\Sigma A, C)$. This leads to interesting structure, describing (with upper bounds!) how trivially a particular sequence of arrows may compose.

To begin, consider a sequence of maps $A_0 \to A_1 \to A_2 \to A_3$. If the composites $A_0 \to A_2$ and $A_1\to A_3$ are nulhomotopic, then one has a diagram

\begin{displaymath}
\begin{array}{ccccc}
A_0 & \to & A_1 & \to & * \\
\downarrow & & \downarrow & & \downarrow \\
* & \to & A_2 & \to & A_3
\end{array}
\end{displaymath}
any choice of homotopies in the two squares gives a map $\Sigma A_0 \to A_3$.

\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

Define $C$ and $D$ to be the cofibers of $A_0 \to A_1$ and $A_1\to A_2$, respectively. A choice of homotopy $A_0 \to A_2 \sim 0$ corresponds to a choice of factorization $A_1 \to C \to A_2$, which gives a diagram of pushout squares

\begin{displaymath}
\begin{array}{ccccccc}
A_0 & \to & A_1 & \to & * \\
\downarrow & & \downarrow & & \downarrow \\
* & \to & C &\to & \Sigma A_0  & \to & *\\
 & & \downarrow & & \downarrow & & \downarrow \\
 & & A_2 & \to & D & \to & C'
\end{array}
\end{displaymath}
It is to be noted that the map $\Sigma A_0 \to D$ and possibly the object $C'$ depend on the choice of factor $C \to A_2$, but that $A_2 \to D$ does not, in any meaningful sense, so depend: this is just the structure map of the cofiber of $A_1\to A_2$. Note that the cofiber $C'$ of $C\to A_2$ is thus equivalent to that of $\Sigma A_0 \to D$; but again the role of choices must be studied.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A sequence of maps $A_0 \to A_1 \to \cdots \to A_n$ will be called \emph{a bracket sequence} (a novel phrase for the purposes of this entry) in either of two cases:

\begin{itemize}%
\item $n = 3$ and the composites $A_0 \to A_2$ and $A_1 \to A_3$ are nulhomotopic; OR
\item $n \gt 3$, and (using the preceding notations), there are choices of factor $C\to A_2$ and $D \to A_3$ such that the induced sequence $\Sigma A_0 \to D \to A_3 \to \cdots \to A_n$ is a bracket sequence.

\end{itemize}
In all cases, a bracket sequence leads to a three-map sequence

\begin{displaymath}
\Sigma^m A_0 \to D_m \to A_{m+2} \to A_{m+3}
\end{displaymath}
in which consecutive maps compose trivially, and so there are induced choices of maps

\begin{displaymath}
\Sigma^{m+1} A_0 \to A_{m+3} .
\end{displaymath}
The collection of all such maps, taking all compatible variations, is the \textbf{Toda Bracket} of the bracket sequence.

Among the bracket sequences, a particular family arises which here will be called \emph{null-bracket} (again, a novel phrase). A sequence will be called null-bracket if

\begin{itemize}%
\item $n=2$ and $A_0 \to A_2$ is trivial, OR
\item $n \gt 2$, and there is a choice of factorization $A_1 \to C \to A_2$ such that the sequence $C \to A_2 \to \cdots \to A_n$ is null-bracket.

\end{itemize}
If the Toda bracket for a bracket sequence includes the trivial map $\Sigma^{m+1} A_0 \to A_{m+3}$ then the sequence is null-bracket.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

By definition, if a sequence is a bracket sequence AND NOT a null-bracket sequence, it follows that all the relevant maps $\Sigma^{k} A_0 \to A_n$ are nontrivial. Things like these Toda brackets have been studied by many \emph{(FIXME: referrences later)} and especially the length-three brackets used by H. Toda to describe most of $\pi_k \mathbb{S}^n$ for $k \lt 31$ or so.

In (\hyperlink{Cohen}{Cohen, 1968}) is given a criterion for stable maps of spheres to inhabit non-null Toda brackets; this turns out to be most of $\pi_* \mathbb{S}$, and furthermore the maps in the bracket sequences can be chosen from a very small set (\_FIXME\_: be more precise! degree maps $n \iota$, [[Hopf map]]s $\eta, \theta,\sigma$, and $\alpha_p$\ldots{} )

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Massey product]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Joel Cohen, \emph{The decomposition of stable homotopy}, Annals of Mathematics (2) 87 (2): 305--320 (1968)

\end{itemize}
\begin{itemize}%
\item [[Stanley Kochmann]], section 5.7 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Hans-Joachim Baues]], \emph{On the cohomology of categories, universal Toda brackets and homotopy pairs}, K-Theory \textbf{11}:3, April 1997, pp. 259-285 (27) \href{http://link.springer.com/article/10.1023/A%3A1007796409912}{springer}


\item Boryana Dimitrova, \emph{Universal Toda brackets of commutative ring spectra}, poster, Bonn 2010, \href{http://www.math.uni-bonn.de/people/grk1150/YWT2010/YWT_Dimitrova.pdf}{pdf}


\item C. Roitzheim, [[S. Whitehouse]], \emph{Uniqueness of $A_\infty$-structures and Hochschild cohomology}, \href{http://arxiv.org/abs/0909.3222}{arxiv/0909.3222}


\item [[Steffen Sagave]], \emph{Universal Toda brackets of ring spectra}, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008, \href{http://arxiv.org/abs/math/0611808}{math.KT/0611808}



\end{itemize}
[[!redirects Toda Brackets]]

[[!redirects Toda brackets]]



\end{document}